Loewner chains and parametric representation of biholomorphic mappings in complex Banach spaces

Let X be a complex Banach space and let B be the unit ball of X. In this paper we obtain sufficient conditions for biholomorphic mappings on B to have parametric representation. Also we study certain properties of Loewner chains, and we obtain infinite dimensional versions of some well known univalence criteria on the unit ball of

Loewner PDE in infinite dimensions

In this paper, we prove the existence and uniqueness of the solution $f(z,t)$ of the Loewner PDE with normalization $Df(0,t)=e^{tA}$, where $A\in L(X,X)$ is such that $k_+(A)<2m(A)$, on the unit ball of a separable reflexive complex Banach space $X$. We also give improvements of the results obtained recently by Hamada and Kohr, but we omit their proofs for the sake of brevity. In particular, we obtain the biholomorphicity of the univalent Schwarz mappings $v(z,s,t)$ with normalization $Dv(0,s,t)=e^{-(t-s)A}$ for $t\geq s\geq 0$, where $m(A)>0$, which satisfy the semigroup property on the unit ball of a complex Banach space $X$. We further obtain the biholomorphicity of $A$-normalized univalent subordination chains under some normality condition on the unit ball of a reflexive complex Banach space $X$. We prove the existence of the biholomorphic solutions $f(z,t)$ of the Loewner PDE with normalization $Df(0,t)=e^{tA}$ on the unit ball of a separable reflexive complex Banach space $X$. The results obtained in this paper give some positive answers to the open problems and conjectures proposed by the authors in 2013

Convex mappings in several complex variables

Let B be the unit ball of Cn with respect to an arbitrary norm. We will give a sufficient condition for a local diffeomorphism of C1 class on B to be univalent and to have a convex image. Finally, we present an application on the complex ellipsoid B(p1, ... , pn), where p1, ... , pn ≥ 1

On strongly starlikeness of order α in several complex variables

In this paper we introduce the concept of strongly starlikeness of order α > 0, for holomorphic mappings defined on the unit ball of Cn. We obtain the distorsion and the covering theorems for strongly starlike mappings of order α ∈ (0,1] and we give a connection between strongly starlikeness and spirallikeness in Cn

Starlike mappings of order alpha on the unit ball in complex Banach spaces

In this paper, we will give the growth theorem of starlike mappings of order α on the unit ball B in complex Banach spaces. We also give an analytic sufficient condition for a locally biholomorphic mapping on B to be a starlike mapping of order α

Convex subordination chains and injective mappings in Cn

AbstractIn this paper we continue the work related to convex subordination chains in C and Cn, and prove that if f(z)=z+∑k=2∞Ak(zk) is a holomorphic mapping on the Euclidean unit ball Bn in Cn such that ∑k=2∞k2‖Ak‖⩽1, a:[0,1]→[0,∞) is a function of class C2 on (0,1) and continuous on [0,1], such that a(1)=0, a(t)>0, ta′(t)>−1/2 for t∈(0,1), and if a(⋅) satisfies a differential equation on (0,1), then f(z,t)=a(t2)Df(tz)(tz)+f(tz) is a convex subordination chain over (0,1] and the mapping F(z)=a(‖z‖2)Df(z)(z)+f(z) is injective on Bn. We also present certain coefficient bounds which provide sufficient conditions for univalence, quasiregularity and starlikeness for the chain f(z,t). Finally we give some examples of convex subordination chains over (0,1]

Certain partial differential subordinations on some Reinhardt domains in Cnℂ^n

We obtain an extension of Jack-Miller-Mocanu's Lemma for holomorphic mappings defined in some Reinhardt domains in $ℂ^n$. Using this result we consider first and second order partial differential subordinations for holomorphic mappings defined on the Reinhardt domain $B_{2p}$ with p ≥ 1